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Theorem hosval 25323
Description: Value of the sum of two Hilbert space operators. (Contributed by NM, 10-Nov-2000.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Assertion
Ref Expression
hosval  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )

Proof of Theorem hosval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hosmval 25318 . . . 4  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( S  +op  T
)  =  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) )
21fveq1d 5804 . . 3  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) ) `
 A ) )
3 fveq2 5802 . . . . 5  |-  ( x  =  A  ->  ( S `  x )  =  ( S `  A ) )
4 fveq2 5802 . . . . 5  |-  ( x  =  A  ->  ( T `  x )  =  ( T `  A ) )
53, 4oveq12d 6221 . . . 4  |-  ( x  =  A  ->  (
( S `  x
)  +h  ( T `
 x ) )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
6 eqid 2454 . . . 4  |-  ( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x ) ) )  =  ( x  e. 
~H  |->  ( ( S `
 x )  +h  ( T `  x
) ) )
7 ovex 6228 . . . 4  |-  ( ( S `  A )  +h  ( T `  A ) )  e. 
_V
85, 6, 7fvmpt 5886 . . 3  |-  ( A  e.  ~H  ->  (
( x  e.  ~H  |->  ( ( S `  x )  +h  ( T `  x )
) ) `  A
)  =  ( ( S `  A )  +h  ( T `  A ) ) )
92, 8sylan9eq 2515 . 2  |-  ( ( ( S : ~H --> ~H  /\  T : ~H --> ~H )  /\  A  e. 
~H )  ->  (
( S  +op  T
) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
1093impa 1183 1  |-  ( ( S : ~H --> ~H  /\  T : ~H --> ~H  /\  A  e.  ~H )  ->  ( ( S  +op  T ) `  A )  =  ( ( S `
 A )  +h  ( T `  A
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    |-> cmpt 4461   -->wf 5525   ` cfv 5529  (class class class)co 6203   ~Hchil 24500    +h cva 24501    +op chos 24519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-hilex 24580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-hosum 25313
This theorem is referenced by:  hoscl  25328  hoaddcomi  25355  hodsi  25358  hoaddassi  25359  hocadddiri  25362  hoaddid1i  25369  honegsubi  25379  hoadddi  25386  hoadddir  25387  lnophsi  25584  hmops  25603  adjadd  25676  nmoptrii  25677  leopadd  25715  pjsdii  25738  pjscji  25753  pjtoi  25762
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